1
00:01:03 --> 00:01:09
Okay, that's,
so to speak,

2
00:01:05 --> 00:01:11
the text for today.
The Fourier series,

3
00:01:09 --> 00:01:15
and the Fourier expansion for f
of t,

4
00:01:14 --> 00:01:20
so f of t, if it looks like
this should be periodic,

5
00:01:19 --> 00:01:25
and two pi should be a period.
Sometimes people rather

6
00:01:25 --> 00:01:31
sloppily say periodic with
period two pi,

7
00:01:29 --> 00:01:35
but that's a little ambiguous.
So, this period could also be

8
00:01:37 --> 00:01:43
pi or a half pi or something
like that as well.

9
00:01:42 --> 00:01:48
The an's and bn's are
calculated according to these

10
00:01:47 --> 00:01:53
formulas.
Now, we're going to need in

11
00:01:51 --> 00:01:57
just a minute a consequence of
those formulas,

12
00:01:56 --> 00:02:02
which, it's not subtle,
but because there are formulas

13
00:02:01 --> 00:02:07
for an and bn,
it follows that once you know f

14
00:02:06 --> 00:02:12
of t,
the an's and bn's are

15
00:02:10 --> 00:02:16
determined.
Or, to put it another way,

16
00:02:15 --> 00:02:21
a function cannot have two
different Fourier series.

17
00:02:20 --> 00:02:26
Or, to put it yet another way,
if f of t,

18
00:02:24 --> 00:02:30
if two functions are equal,
you'll see why I write it in

19
00:02:30 --> 00:02:36
this rather peculiar form.
Then, the Fourier series for f

20
00:02:35 --> 00:02:41
is the same as the Fourier
series for g.

21
00:02:40 --> 00:02:46
And, the reason is because if f
is equal to g,

22
00:02:44 --> 00:02:50
then this integral with an f
there is the same as the

23
00:02:49 --> 00:02:55
integral with a g there.
And therefore,

24
00:02:52 --> 00:02:58
the an's come out to be the
same.

25
00:02:55 --> 00:03:01
In the same way,
the bn's come out to be the

26
00:02:58 --> 00:03:04
same.
So, the Fourier series are the

27
00:03:01 --> 00:03:07
same, coefficient by
coefficient, for f and g.

28
00:03:05 --> 00:03:11
Now, my ultimate goal-- let's
all put down the argument since

29
00:03:10 --> 00:03:16
there are formulas,
since we have formulas for an

30
00:03:14 --> 00:03:20
and bn.
Now, a consequence of that is,

31
00:03:19 --> 00:03:25
well, let me first say,
what I'm aiming at is you will

32
00:03:23 --> 00:03:29
be amazed at how long it's going
to take me to get to this.

33
00:03:29 --> 00:03:35
I just want to calculate the
Fourier series for some rather

34
00:03:33 --> 00:03:39
simple periodic function.
It's going to look like this.

35
00:03:38 --> 00:03:44
So, here's pi,
and here's negative pi.

36
00:03:41 --> 00:03:47
So, the function which just
looks like t in between those

37
00:03:45 --> 00:03:51
two, so, it goes up to,
it's a function,

38
00:03:49 --> 00:03:55
t, more or less,
goes up to pi here,

39
00:03:51 --> 00:03:57
minus pi there.
But, of course,

40
00:03:54 --> 00:04:00
it's got to be periodic of
period two pi.

41
00:03:59 --> 00:04:05
Well, then, it just repeats
itself after that.

42
00:04:02 --> 00:04:08
After this, it just does that,
and so on.

43
00:04:04 --> 00:04:10
It's a little ambiguous what
happens at these endpoints.

44
00:04:08 --> 00:04:14
Well, let's not worry about
that for the moment,

45
00:04:12 --> 00:04:18
and frankly,
it won't really matter because

46
00:04:15 --> 00:04:21
the integrals don't care about
what happens in individual

47
00:04:19 --> 00:04:25
points.
So, there's my f of t.

48
00:04:21 --> 00:04:27
Now, I, of course,
could start doing it right

49
00:04:24 --> 00:04:30
away.
But, you will quickly find,

50
00:04:27 --> 00:04:33
if you start doing these
problems and hacking around with

51
00:04:30 --> 00:04:36
them, that the calculations seem
really quite long.

52
00:04:34 --> 00:04:40
And therefore,
in the first half of the

53
00:04:37 --> 00:04:43
period, the first half of the
period I want to show you how to

54
00:04:41 --> 00:04:47
shorten the calculations.
And in the second half of the

55
00:04:47 --> 00:04:53
period, after we've done that
and calculated this thing

56
00:04:50 --> 00:04:56
successfully,
I hope, I want to show you how

57
00:04:54 --> 00:05:00
to remove various restrictions
on these functions,

58
00:04:57 --> 00:05:03
how to extend the range of
Fourier series.

59
00:05:01 --> 00:05:07
Well, one obvious thing,
for example,

60
00:05:03 --> 00:05:09
is suppose the function isn't
periodic of period two pi.

61
00:05:06 --> 00:05:12
Suppose it has some other
period.

62
00:05:08 --> 00:05:14
Does that mean there's no
formula?

63
00:05:10 --> 00:05:16
Well, of course not.
There's a formula.

64
00:05:13 --> 00:05:19
But, we need to know what it
is, particularly in the

65
00:05:16 --> 00:05:22
applications,
the period is rarely two pi.

66
00:05:19 --> 00:05:25
It's normally one,
or something like that.

67
00:05:21 --> 00:05:27
But, let's first of all,
I'm sure what you will

68
00:05:24 --> 00:05:30
appreciate is how the
calculations can get shortened.

69
00:05:29 --> 00:05:35
Now, the main way of shortening
them is by using evenness and

70
00:05:35 --> 00:05:41
oddness.
And, what I claim is this,

71
00:05:39 --> 00:05:45
that if f of t is an
even function,

72
00:05:44 --> 00:05:50
remember what that means,
that f of negative t is equal

73
00:05:51 --> 00:05:57
to f of t.
Cosine is a good example,

74
00:05:57 --> 00:06:03
of course, cosine nt;
are all these

75
00:06:02 --> 00:06:08
functions are even functions.
If f of t is even,

76
00:06:07 --> 00:06:13
then its Fourier series
contains only the cosine terms.

77
00:06:16 --> 00:06:22
In other words,
half the calculations you don't

78
00:06:21 --> 00:06:27
have to do if you start with an
even function.

79
00:06:26 --> 00:06:32
That's what I mean by
shortening the work.

80
00:06:31 --> 00:06:37
There are no odd terms,
or let's put it positively.

81
00:06:37 --> 00:06:43
All the bn's are zero.
Now, one way of doing this

82
00:06:42 --> 00:06:48
would be to say,
well, y to the bn zero,

83
00:06:44 --> 00:06:50
well, we've got formulas,
and fool around with the

84
00:06:47 --> 00:06:53
formula for the bn,
and think about a little bit,

85
00:06:50 --> 00:06:56
and finally decide that that
has to come out to be zero.

86
00:06:53 --> 00:06:59
That's not a bad way,
and it would remind you of some

87
00:06:56 --> 00:07:02
basic facts about integration,
about integrals.

88
00:07:00 --> 00:07:06
Instead of doing that,
I'm going to apply my little

89
00:07:04 --> 00:07:10
principle that if two functions
are the same,

90
00:07:08 --> 00:07:14
then their Fourier series have
to be the same.

91
00:07:12 --> 00:07:18
So, the argument I'm going to
give is this,

92
00:07:16 --> 00:07:22
so, I'm going to try to prove
this statement now.

93
00:07:20 --> 00:07:26
And, I'm going to use the facts
on the first board to do it.

94
00:07:25 --> 00:07:31
So, what is f of minus t?

95
00:07:30 --> 00:07:36
Well, if that's equal to f of
t, then in terms of the

96
00:07:35 --> 00:07:41
Fourier series,
how do I get the Fourier series

97
00:07:39 --> 00:07:45
for f of minus t?
Well, I take the Fourier series

98
00:07:44 --> 00:07:50
for f of t, and substitute t
equals minus t.

99
00:07:48 --> 00:07:54
Now, what happens when I do
that?

100
00:07:51 --> 00:07:57
So, the Fourier series for this
looks like a zero over two

101
00:07:56 --> 00:08:02
plus summation what?
Well, the an cosine nt,

102
00:08:02 --> 00:08:08
that does not change
because when I change t to

103
00:08:06 --> 00:08:12
negative t,
the cosine nt does

104
00:08:11 --> 00:08:17
not change, stays the same
because it's an even function.

105
00:08:15 --> 00:08:21
What happens to the sine term?
Well, the sine of negative nt

106
00:08:20 --> 00:08:26
is equal to minus the
sine of nt.

107
00:08:25 --> 00:08:31
So, the other terms,
the sine terms change sign.

108
00:08:30 --> 00:08:36
So, all that's the result of
substituting t for negative t

109
00:08:34 --> 00:08:40
and f of t.

110
00:08:36 --> 00:08:42
On the other hand,
what's f of t itself?

111
00:08:40 --> 00:08:46
Well, f of t itself is what
happened before that.

112
00:08:43 --> 00:08:49
Now it's got a plus sign
because nothing was done to the

113
00:08:48 --> 00:08:54
series.
Well, if the function is even,

114
00:08:50 --> 00:08:56
then those two right hand sides
are the same function.

115
00:08:54 --> 00:09:00
In other words,
they're like my f of t equals g

116
00:08:58 --> 00:09:04
of t. And therefore,

117
00:09:02 --> 00:09:08
the Fourier series on the left
must be the same.

118
00:09:06 --> 00:09:12
In other words,
if these are equal,

119
00:09:09 --> 00:09:15
therefore, these have to be
equal, too.

120
00:09:13 --> 00:09:19
Now, there's no problem with
the cosine terms.

121
00:09:17 --> 00:09:23
They look the same.
On the other hand,

122
00:09:20 --> 00:09:26
the sine terms have changed
sign.

123
00:09:23 --> 00:09:29
Therefore, it must be the case
that bn is always equal to

124
00:09:28 --> 00:09:34
negative bn for all n.
That's the only way this series

125
00:09:34 --> 00:09:40
can be the same as that one.
Now, if bn is equal to negative

126
00:09:39 --> 00:09:45
bn,
that implies that bn is zero.

127
00:09:43 --> 00:09:49
Zero is the only number which

128
00:09:46 --> 00:09:52
is equal to its negative.
And so, by this argument,

129
00:09:51 --> 00:09:57
in other words,
using the uniqueness of Fourier

130
00:09:54 --> 00:10:00
series, we conclude that if the
function is even,

131
00:09:59 --> 00:10:05
then its Fourier series can
only have cosine terms in it.

132
00:10:05 --> 00:10:11
Now, you say,
hey, that's obvious.

133
00:10:07 --> 00:10:13
The cosine, that's just a point
of logic.

134
00:10:09 --> 00:10:15
But, this is a mathematics
course, after all.

135
00:10:12 --> 00:10:18
It's not just about
calculation.

136
00:10:14 --> 00:10:20
Many of you would say,
yeah, of course that's obvious

137
00:10:18 --> 00:10:24
because cosines are even,
and the sines are odd.

138
00:10:21 --> 00:10:27
I say, yeah,
and so why does that make it

139
00:10:24 --> 00:10:30
true?
Well, the cosine's even.

140
00:10:25 --> 00:10:31
Plus t into minus t,
and what you are proving

141
00:10:29 --> 00:10:35
is the converse.
The converse is obvious.

142
00:10:33 --> 00:10:39
Yeah, obvious,
I don't care.

143
00:10:35 --> 00:10:41
If the right-hand side is the
sum of the functions,

144
00:10:39 --> 00:10:45
well, so is the left.
But I'm saying it the other way

145
00:10:43 --> 00:10:49
around.
If the left is an even

146
00:10:45 --> 00:10:51
function, why does the
right-hand side have to have

147
00:10:49 --> 00:10:55
only even terms in it?
And, this is the argument which

148
00:10:53 --> 00:10:59
makes that true.
Now, there is a further

149
00:10:56 --> 00:11:02
simplification because if you've
got an even function,

150
00:11:00 --> 00:11:06
oh, by the way,
of course the same thing is

151
00:11:03 --> 00:11:09
true for the odd,
I ought to put that down,

152
00:11:06 --> 00:11:12
and so also,
if f of t is odd,

153
00:11:09 --> 00:11:15
then I think one of these
proofs is enough.

154
00:11:14 --> 00:11:20
The other you can supply
yourself.

155
00:11:17 --> 00:11:23
That will imply that all the
an's are zero,

156
00:11:20 --> 00:11:26
even including this first one,
a zero,

157
00:11:25 --> 00:11:31
and by the same reasoning.

158
00:11:28 --> 00:11:34

159
00:11:37 --> 00:11:43
So, an even function uses only
cosines for its Fourier

160
00:11:41 --> 00:11:47
expansion.
An odd function uses only

161
00:11:44 --> 00:11:50
sines.
Good.

162
00:11:45 --> 00:11:51
But, we still have to,
suppose we got an even

163
00:11:49 --> 00:11:55
function.
We've still got to calculate

164
00:11:53 --> 00:11:59
this integral.
Well, even that can be

165
00:11:56 --> 00:12:02
simplified.
So, the second stage of the

166
00:11:59 --> 00:12:05
simplification,
again, assuming that we have an

167
00:12:04 --> 00:12:10
even or odd function,
and by the way,

168
00:12:07 --> 00:12:13
[LAUGHTER].
Totally unauthorized.

169
00:12:11 --> 00:12:17

170
00:12:26 --> 00:12:32
So, if f of t is even,
what we'd like to do now is

171
00:12:34 --> 00:12:40
simplify the integral a little.
And, there is an easy way to do

172
00:12:43 --> 00:12:49
that, because,
look, if f of t is an even

173
00:12:49 --> 00:12:55
function, then so is f of t
cosine nt,

174
00:12:57 --> 00:13:03
is also even.
Imagine, we could make little

175
00:13:02 --> 00:13:08
rules about an even function
times an even function is an

176
00:13:06 --> 00:13:12
even function.
There are general rules of that

177
00:13:09 --> 00:13:15
type, and some of you know them,
and they are very useful.

178
00:13:13 --> 00:13:19
But, let's just do it ad hoc
here.

179
00:13:15 --> 00:13:21
If I change t to negative
t here,

180
00:13:18 --> 00:13:24
I don't change the function
because it's even.

181
00:13:21 --> 00:13:27
And, I don't change the cosine
because that's even.

182
00:13:24 --> 00:13:30
So, if I change t to negative
t, I don't change the function.

183
00:13:28 --> 00:13:34
Either factor that function,
and therefore I don't change

184
00:13:32 --> 00:13:38
the product of those two things
either.

185
00:13:36 --> 00:13:42
So, it's also even.
Now, what about an even

186
00:13:41 --> 00:13:47
function when you integrate it?
Here's a typical looking even

187
00:13:48 --> 00:13:54
function, let's say,
something like,

188
00:13:52 --> 00:13:58
I don't know,
wiggle, wiggle,

189
00:13:56 --> 00:14:02
again.
Here's our better even

190
00:13:59 --> 00:14:05
function.
All right, so,

191
00:14:02 --> 00:14:08
minus pi to pi,
even, even though the t-axis is

192
00:14:08 --> 00:14:14
somewhat curvy.
So, there is an even function.

193
00:14:14 --> 00:14:20
The point is that if you
integrate an even function from

194
00:14:17 --> 00:14:23
negative pi to pi,
I think you all know even from

195
00:14:21 --> 00:14:27
calculus you were taught to do
this simplification.

196
00:14:24 --> 00:14:30
Don't do that.
Instead, integrate from zero to

197
00:14:27 --> 00:14:33
pi, and double the answer.
Why should you do that?

198
00:14:31 --> 00:14:37
The answer is because it's
always nice to have zero as one

199
00:14:35 --> 00:14:41
of the limits of integration.
I trust to your experience,

200
00:14:39 --> 00:14:45
I don't have to sell that.
Minus pi is a particularly

201
00:14:43 --> 00:14:49
unpleasant lower limit of
integration because you are sure

202
00:14:47 --> 00:14:53
to get in trouble with negative
signs.

203
00:14:50 --> 00:14:56
There are bound to be at least
three negative signs floating

204
00:14:54 --> 00:15:00
around.
And, if you miss one of them,

205
00:14:57 --> 00:15:03
you'll get the wrong signs of
answer.

206
00:15:01 --> 00:15:07
The answer will have the wrong
sign.

207
00:15:03 --> 00:15:09
So, the way the formula from
this simplifies is that an,

208
00:15:08 --> 00:15:14
instead of integrating from
negative pi to pi,

209
00:15:12 --> 00:15:18
I can integrate only from zero
to pi, and double the answer.

210
00:15:17 --> 00:15:23
So, our better formula is this.
If the function is even,

211
00:15:22 --> 00:15:28
this is the formula you should
use: zero to pi,

212
00:15:26 --> 00:15:32
f of t cosine nt dt.

213
00:15:31 --> 00:15:37
Of course, I don't have to tell

214
00:15:35 --> 00:15:41
you what bn should be because bn
will be zero.

215
00:15:39 --> 00:15:45
And, in the same way,
if f is odd,

216
00:15:42 --> 00:15:48
the same reasoning shows that
bn-- of course,

217
00:15:45 --> 00:15:51
an will be zero this time.
But it will be bn that will be

218
00:15:50 --> 00:15:56
two over pi times the integral
from zero to pi of f of t sine

219
00:15:55 --> 00:16:01
nt dt.

220
00:16:00 --> 00:16:06
Maybe we'd better just a word

221
00:16:03 --> 00:16:09
about that since,
why is that so?

222
00:16:06 --> 00:16:12
If it's odd,
doesn't that mean things become

223
00:16:08 --> 00:16:14
zero?
If you integrate an odd

224
00:16:10 --> 00:16:16
function like that,
the integral over minus pi to

225
00:16:14 --> 00:16:20
pi, you get zero.
Well, but this is not an odd

226
00:16:17 --> 00:16:23
function.
This is an odd function,

227
00:16:19 --> 00:16:25
and this is an odd function.
But the product of two odd

228
00:16:22 --> 00:16:28
functions is an even function.
Odd times odd is even.

229
00:16:26 --> 00:16:32
I said I wasn't going to give
you those rules,

230
00:16:29 --> 00:16:35
but since this is the one which
trips everybody up,

231
00:16:32 --> 00:16:38
maybe I'd better say it just
justbecause it looks wrong.

232
00:16:38 --> 00:16:44
Right, this is odd.
That's odd.

233
00:16:40 --> 00:16:46
Think about it.
If I change t to negative t,

234
00:16:43 --> 00:16:49
this multiplies by
minus one.

235
00:16:46 --> 00:16:52
This multiplies by minus one.
And therefore,

236
00:16:49 --> 00:16:55
the product multiplies by minus
one times minus one.

237
00:16:54 --> 00:17:00
In other words,
it multiplies by plus one.

238
00:16:57 --> 00:17:03
Nothing happens,
so it stays the same.

239
00:17:01 --> 00:17:07
Why does nobody believe this,
even though it's true?

240
00:17:04 --> 00:17:10
It's because they are thinking
about numbers.

241
00:17:08 --> 00:17:14
Everybody knows that an odd
number times an odd number is an

242
00:17:12 --> 00:17:18
odd number.
So, I'm not multiplying numbers

243
00:17:15 --> 00:17:21
here, which also I'll put them
in boxes to indicate that they

244
00:17:20 --> 00:17:26
are not numbers.
How's that?

245
00:17:22 --> 00:17:28
Brand-new invented notation.
The box means caution.

246
00:17:25 --> 00:17:31
The inside is not a number,
it's the word odd or even.

247
00:17:31 --> 00:17:37
It's just a symbolic statement
that the product of an odd

248
00:17:35 --> 00:17:41
function and an odd function is
an even function.

249
00:17:39 --> 00:17:45
Even times even is even.
What's odd times even?

250
00:17:43 --> 00:17:49
Yes, it has to get equal time.
Obviously, something must come

251
00:17:47 --> 00:17:53
out to be odd,
right.

252
00:17:49 --> 00:17:55
Okay, so, now that we've got
our two simplifications,

253
00:17:53 --> 00:17:59
we are ready to do this
problem.

254
00:17:56 --> 00:18:02
Instead of attacking it with
the original formulas,

255
00:18:00 --> 00:18:06
we are going to think about it
and attack it with our better

256
00:18:04 --> 00:18:10
formulas.
So, now we are going to

257
00:18:11 --> 00:18:17
calculate the Fourier series for
f of t.

258
00:18:19 --> 00:18:25
The first thing I see,
so f of t is our little thing

259
00:18:29 --> 00:18:35
here.
Well, first of all,

260
00:18:32 --> 00:18:38
what kind of function is it:
odd, even, or neither?

261
00:18:35 --> 00:18:41
Most functions are neither,
of course.

262
00:18:38 --> 00:18:44
But, fortunately in the
applications,

263
00:18:40 --> 00:18:46
functions tend to be one or the
other.

264
00:18:42 --> 00:18:48
Or, they can be converted into
one to the other.

265
00:18:46 --> 00:18:52
Maybe if I get a chance,
I'll show you a little how,

266
00:18:49 --> 00:18:55
or the recitations will.
So, this function is odd.

267
00:18:52 --> 00:18:58
Okay, half the work just
disappeared.

268
00:18:55 --> 00:19:01
I don't have to calculate any
an's.

269
00:18:57 --> 00:19:03
They will be zero.
So, I only have to calculate

270
00:19:01 --> 00:19:07
bn, and I'll calculate them by
my better formula.

271
00:19:04 --> 00:19:10
So, it's two over pi times the
integral from zero to pi,

272
00:19:08 --> 00:19:14
and what I have to integrate,
well, now, finally you've got

273
00:19:11 --> 00:19:17
to integrate something.
From zero to pi,

274
00:19:14 --> 00:19:20
this is the function,
t.

275
00:19:15 --> 00:19:21
So, I have to integrate t times
sine of nt dt.

276
00:19:18 --> 00:19:24
Okay,

277
00:19:22 --> 00:19:28
so this is why you learned
integration by parts,

278
00:19:25 --> 00:19:31
one of many reasons why you
learned integration by parts,

279
00:19:29 --> 00:19:35
so that you wouldn't have to
pull out your little calculators

280
00:19:32 --> 00:19:38
to do this.
Okay, now, let's do it.

281
00:19:36 --> 00:19:42
So, it's two over pi.

282
00:19:39 --> 00:19:45
Let's solve that away so we can
forget about it.

283
00:19:42 --> 00:19:48
And, what's then left is just
the evaluation of the integral

284
00:19:47 --> 00:19:53
between limits.
So, if I integrate by parts,

285
00:19:50 --> 00:19:56
I'll want to differentiate the
t, and integrate the sign,

286
00:19:54 --> 00:20:00
right?
So, the first step is you don't

287
00:19:57 --> 00:20:03
do the differentiation.
You only do the integration.

288
00:20:02 --> 00:20:08
So, that integrates to be
cosine nt over n,

289
00:20:05 --> 00:20:11
more or less.
The only thing is,

290
00:20:08 --> 00:20:14
if I differentiate this,
I get negative sine nt

291
00:20:11 --> 00:20:17
instead of,
so, I want to put a negative

292
00:20:15 --> 00:20:21
sign in front of all this.
And, I will evaluate that

293
00:20:19 --> 00:20:25
between the limits,
zero and pi,

294
00:20:21 --> 00:20:27
and then subtract what you get
by doing both things,

295
00:20:25 --> 00:20:31
both the differentiation and
the integration.

296
00:20:28 --> 00:20:34
So, I subtract the integral
from zero to pi.

297
00:20:33 --> 00:20:39
I now differentiate the t,
and integrate.

298
00:20:35 --> 00:20:41
Well, I just did the
integration.

299
00:20:37 --> 00:20:43
That's negative cosine nt over
n.

300
00:20:40 --> 00:20:46
You see how the negative signs
pile up?

301
00:20:43 --> 00:20:49
And, if this is negative pi
instead of zero,

302
00:20:45 --> 00:20:51
it's at that point when it
starts to lose heart.

303
00:20:48 --> 00:20:54
You see three negative signs,
and then when you substitute,

304
00:20:52 --> 00:20:58
you're going to have to put in
still something else negative,

305
00:20:56 --> 00:21:02
and you just have the feeling
you're going to make a mistake.

306
00:21:01 --> 00:21:07
And, you will.
Okay, now all we have to do is

307
00:21:05 --> 00:21:11
a little evaluation.
Let's see, at the lower limit I

308
00:21:09 --> 00:21:15
get zero, here.
Let's right away,

309
00:21:12 --> 00:21:18
as two over pi.
At the lower limit,

310
00:21:16 --> 00:21:22
I get zero.
That's nice.

311
00:21:18 --> 00:21:24
At the upper limit,
I get minus pi over n times the

312
00:21:23 --> 00:21:29
cosine of n pi.

313
00:21:26 --> 00:21:32
Now, once and for all,
the cosine of n pi--

314
00:21:31 --> 00:21:37
If you like to make
separate steps out of

315
00:21:35 --> 00:21:41
everything, okay,
I'll let you do it this time,

316
00:21:39 --> 00:21:45
--
-- but in the long run,

317
00:21:43 --> 00:21:49
it's good to remember that
that's negative one to the n'th

318
00:21:48 --> 00:21:54
power
The cosine of pi is minus one .

319
00:21:51 --> 00:21:57
The cosine of two pi is plus
one,

320
00:21:55 --> 00:22:01
three pi, minus one,
and so on.

321
00:22:00 --> 00:22:06
So, at the upper limit,
we get minus pi over n,

322
00:22:05 --> 00:22:11
oh, I didn't finish the
calculation, times the cosine of

323
00:22:11 --> 00:22:17
n pi,
which is minus one to the n'th

324
00:22:17 --> 00:22:23
power.
And now, how about the other

325
00:22:22 --> 00:22:28
guy?
Shall we do in our heads?

326
00:22:26 --> 00:22:32
Well, I can do it in my head,
but I'm not so sure about your

327
00:22:32 --> 00:22:38
heads.
Maybe just this once we won't.

328
00:22:37 --> 00:22:43
What is it?
It's plus sine nt,

329
00:22:41 --> 00:22:47
right?
So, I combined the two negative

330
00:22:44 --> 00:22:50
signs to a plus sign by putting
one this way and the other one

331
00:22:50 --> 00:22:56
that way.
And then, if I integrate that

332
00:22:53 --> 00:22:59
now, it's sine nt divided by n
squared,

333
00:22:58 --> 00:23:04
right?
And that's evaluated between

334
00:23:02 --> 00:23:08
zero and pi.
And of course,

335
00:23:05 --> 00:23:11
the sign function vanishes at
both ends.

336
00:23:09 --> 00:23:15
So, that part is simply zero.
And so, the final answer is

337
00:23:14 --> 00:23:20
that bn is equal to,
well, the pi's cancel.

338
00:23:19 --> 00:23:25
This minus combines with those
n to make one more.

339
00:23:23 --> 00:23:29
And so, the answer is two over
n times minus one to the n plus

340
00:23:30 --> 00:23:36
first power.

341
00:23:35 --> 00:23:41
And therefore,
the final result is that our

342
00:23:40 --> 00:23:46
Fourier series,
the Fourier series for f of t,

343
00:23:46 --> 00:23:52
that funny function
is, the Fourier series is

344
00:23:53 --> 00:23:59
summation bn,
which is two,

345
00:23:56 --> 00:24:02
put the two out front because
it's in every term.

346
00:24:04 --> 00:24:10
There's no reason to repeat it,
minus one to the n plus first

347
00:24:10 --> 00:24:16
power over n times the sign of
nt.

348
00:24:16 --> 00:24:22
That's summed from one to

349
00:24:19 --> 00:24:25
infinity.
Let's stop and take a look at

350
00:24:23 --> 00:24:29
that for a second.
Does that look right?

351
00:24:28 --> 00:24:34
Okay, here's our function.

352
00:24:32 --> 00:24:38

353
00:24:41 --> 00:24:47
Here's our function.
What's the first term of this?

354
00:24:48 --> 00:24:54
When n is one,
this is plus one.

355
00:24:54 --> 00:25:00
So, the first term is sine t.

356
00:25:00 --> 00:25:06
What's the next term?
When n is two,

357
00:25:04 --> 00:25:10
this is negative.
So, it's minus one to the third

358
00:25:08 --> 00:25:14
power.
So, that's negative one over

359
00:25:11 --> 00:25:17
two.
So, it's minus one half sine

360
00:25:13 --> 00:25:19
two t,
and then it obviously continues

361
00:25:17 --> 00:25:23
in the same way plus a third
sign three t.

362
00:25:21 --> 00:25:27
Now, watch carefully because
what I'm going to say in the

363
00:25:24 --> 00:25:30
next minute is the heart of
Fourier series.

364
00:25:27 --> 00:25:33
I've given you that visual to
look at to try to reinforce

365
00:25:31 --> 00:25:37
this, but it's really very
important, as you go to the

366
00:25:34 --> 00:25:40
terminal yourself and do that
work, simple as it is,

367
00:25:38 --> 00:25:44
and pay attention now.
Now, if you think

368
00:25:42 --> 00:25:48
old-fashioned,
i.e.

369
00:25:43 --> 00:25:49
if you think taylor series,
you're not going to believe

370
00:25:47 --> 00:25:53
this because you will say,
well, let's see,

371
00:25:50 --> 00:25:56
these go on and on.
Obviously, it's the first term

372
00:25:53 --> 00:25:59
that's the important one.
That's two sine t.

373
00:25:57 --> 00:26:03
Now, the derivative,
two sine t, sine t would

374
00:26:00 --> 00:26:06
exactly follow the pink curve.
Sine t would look like this.

375
00:26:06 --> 00:26:12
Two sine t goes up
with the wrong angle.

376
00:26:10 --> 00:26:16
The first term,
in other words,

377
00:26:12 --> 00:26:18
does this.
It's going off with the wrong

378
00:26:15 --> 00:26:21
slope.
Now, that's the whole point of

379
00:26:18 --> 00:26:24
Fourier series.
Fourier series is not trying to

380
00:26:22 --> 00:26:28
approximate the function at zero
at the central starting point

381
00:26:27 --> 00:26:33
the way Taylor series do.
Fourier series tries to treat

382
00:26:31 --> 00:26:37
the whole interval,
and approximate the function

383
00:26:35 --> 00:26:41
nicely over the entire interval,
in this case,

384
00:26:38 --> 00:26:44
minus pi to pi,
as well as possible.

385
00:26:40 --> 00:26:46
Taylor series concentrates at
this point, does it the best it

386
00:26:44 --> 00:26:50
can at this point.
Then it tries,

387
00:26:46 --> 00:26:52
with the next term,
to do a little better,

388
00:26:49 --> 00:26:55
and then a little better.
The whole philosophy is

389
00:26:52 --> 00:26:58
entirely different.
Taylor series are used for

390
00:26:55 --> 00:27:01
analyzing what a function of
looks like which you stick close

391
00:26:59 --> 00:27:05
to the base point.
Fourier series analyze what a

392
00:27:04 --> 00:27:10
function looks like over the
whole interval.

393
00:27:07 --> 00:27:13
And, to do that,
you should therefore aim to,

394
00:27:10 --> 00:27:16
so the first approximation is
going to look like that,

395
00:27:14 --> 00:27:20
going to have entirely the
wrong slope.

396
00:27:16 --> 00:27:22
But, the next one will subtract
off something which sort of

397
00:27:21 --> 00:27:27
helps to fix it up.
I can't draw this.

398
00:27:23 --> 00:27:29
That's why I'm sending you to
the visual because the visual

399
00:27:27 --> 00:27:33
draws them beautifully.
And, it shows you how each

400
00:27:31 --> 00:27:37
successive term corrects the
Fourier series,

401
00:27:34 --> 00:27:40
and makes the sum a little
closer to what you started with.

402
00:27:40 --> 00:27:46
So, the next guy would,
let's see, so it's 2t.

403
00:27:44 --> 00:27:50
So, I'm subtracting off,
probably I'm just guessing,

404
00:27:50 --> 00:27:56
but I don't dare draw this.
I haven't prepared to draw it,

405
00:27:56 --> 00:28:02
and I know I'll get it wrong.
So, okay, your exercise.

406
00:28:02 --> 00:28:08
But, it'll look better.
It'll go, maybe,

407
00:28:07 --> 00:28:13
something like,
let's see, it has to end up...

408
00:28:12 --> 00:28:18
some of it gets subtracted
off...

409
00:28:17 --> 00:28:23
I don't know what it looks
like.

410
00:28:20 --> 00:28:26
When you use the visual at the
computer terminal,

411
00:28:25 --> 00:28:31
I've asked you to use it three
times on a variety of functions.

412
00:28:32 --> 00:28:38
I think this is maybe even one
of them.

413
00:28:34 --> 00:28:40
Notice that you can set the
parameter, you can set the

414
00:28:38 --> 00:28:44
coefficients independently.
In other words,

415
00:28:41 --> 00:28:47
you can go back and correct
your works, improving the

416
00:28:45 --> 00:28:51
earlier coefficients,
and it won't affect anything

417
00:28:48 --> 00:28:54
you did before.
But, the most vivid way to do

418
00:28:51 --> 00:28:57
it is to try to get,
visually, by moving the slider,

419
00:28:55 --> 00:29:01
to try to get the very best
value for the first coefficient

420
00:28:59 --> 00:29:05
you can, and look at the curve.
Then get the very best value

421
00:29:05 --> 00:29:11
for the second coefficient and
see how that improves the

422
00:29:09 --> 00:29:15
approximation,
and the third,

423
00:29:11 --> 00:29:17
and so on.
And, the point is,

424
00:29:13 --> 00:29:19
watch the approximations
approaching the function nicely

425
00:29:18 --> 00:29:24
over the whole interval instead
of concentrating all their

426
00:29:22 --> 00:29:28
goodness at the origin the way a
Taylor series would.

427
00:29:26 --> 00:29:32
Now, there is still one
mathematical point left.

428
00:29:30 --> 00:29:36
It's that equality sign,
which is wrong.

429
00:29:35 --> 00:29:41
Why is it wrong?
Well, what I'm saying is that

430
00:29:38 --> 00:29:44
if I add that the series,
it adds up to f of t.

431
00:29:43 --> 00:29:49
Now, it almost does but not
quite.

432
00:29:46 --> 00:29:52
And, I'd better give you the
rule, the theorem.

433
00:29:50 --> 00:29:56
Of all the theorems in this
course that aren't being proved,

434
00:29:56 --> 00:30:02
this is the one that would be
most outside the scope of this

435
00:30:01 --> 00:30:07
course, the one which I would
most like to prove,

436
00:30:05 --> 00:30:11
in fact, just because I'm a
mathematician but wouldn't dare.

437
00:30:12 --> 00:30:18
The theorem tells you when a
Fourier series converges to the

438
00:30:17 --> 00:30:23
function you started with.
And, the essence of it is this.

439
00:30:22 --> 00:30:28
If f is continuous,
is a continuous function,

440
00:30:26 --> 00:30:32
let's give the point,
it's confusing just to keep

441
00:30:30 --> 00:30:36
calling it t.
If you like,

442
00:30:32 --> 00:30:38
call it t, but I think it would
be better to call it t zero

443
00:30:37 --> 00:30:43
just to indicate I'm
looking at a specific point.

444
00:30:44 --> 00:30:50
So, if the function is
continuous there,

445
00:30:48 --> 00:30:54
the value of f of t is
equal to, the Fourier series

446
00:30:54 --> 00:31:00
converges, and it's equal to its
Fourier series,

447
00:30:59 --> 00:31:05
the sum of the Fourier series
at t zero.

448
00:31:05 --> 00:31:11
And, the fact that I can even
use the word sum means that the

449
00:31:09 --> 00:31:15
Fourier series converges.
In other words,

450
00:31:12 --> 00:31:18
when you add up all these guys,
you don't go to infinity or get

451
00:31:16 --> 00:31:22
something which just oscillates
around crazily.

452
00:31:20 --> 00:31:26
They really do add up to
something.

453
00:31:22 --> 00:31:28
Now, if f is not continuous at
t zero,

454
00:31:26 --> 00:31:32
this emphatically will not be
the case.

455
00:31:28 --> 00:31:34
It will definitely not,
but by far, the kinds of

456
00:31:32 --> 00:31:38
discontinuities which occur in
the applications are ones like

457
00:31:36 --> 00:31:42
in this picture,
where the discontinuities are

458
00:31:39 --> 00:31:45
jump discontinuities.
They are almost always jump

459
00:31:44 --> 00:31:50
discontinuities.
And, in that case,

460
00:31:47 --> 00:31:53
in other words,
they are isolated.

461
00:31:49 --> 00:31:55
The function looks good here
and here, but there's a break.

462
00:31:53 --> 00:31:59
Typically, electrical engineers
just don't leave a gap because

463
00:31:57 --> 00:32:03
they like, I don't know why.
But electrical engineer,

464
00:32:02 --> 00:32:08
and others of his or her ilk
would draw that function like

465
00:32:09 --> 00:32:15
this, like a rip saw tooth.
Even those vertical lines have

466
00:32:16 --> 00:32:22
no meaning whatever,
but they make people look

467
00:32:21 --> 00:32:27
happier.
So, if f has a jump

468
00:32:24 --> 00:32:30
discontinuity at t zero,
and as I said,

469
00:32:30 --> 00:32:36
that's the most important kind,
then f of t,

470
00:32:36 --> 00:32:42
then the Fourier series adds up
to, converges to,

471
00:32:42 --> 00:32:48
it converges,
and it converges to the mid

472
00:32:46 --> 00:32:52
point of the jump.
Let me just write it out in

473
00:32:52 --> 00:32:58
words like that,
the midpoint of the jump.

474
00:32:55 --> 00:33:01
That's the way we'll be using
it in this course.

475
00:32:58 --> 00:33:04
There's a notation for this,
and it's in your book.

476
00:33:02 --> 00:33:08
But, those of you who would be
interested in such things would

477
00:33:07 --> 00:33:13
know it anyway.
So, let's just call it the

478
00:33:11 --> 00:33:17
midpoint of the jump.
So, if I ask you,

479
00:33:14 --> 00:33:20
to what does this converge?
In other words,

480
00:33:18 --> 00:33:24
this series,
what this shows is that the

481
00:33:22 --> 00:33:28
series, I'll write it out in the
abbreviated form,

482
00:33:26 --> 00:33:32
summation minus one to the n
plus one over n sine nt,

483
00:33:32 --> 00:33:38
what's the sum of the series?

484
00:33:39 --> 00:33:45
What is it?
Let's call this not 

485
00:33:41 --> 00:33:47
little f of t.
Let's call it capital F of t.

486
00:33:44 --> 00:33:50
I want to know,

487
00:33:46 --> 00:33:52
what's the graph of capital F
of t?

488
00:33:48 --> 00:33:54
Well, the initial thing is to
say, well, it must be the same

489
00:33:53 --> 00:33:59
as the graph of the function you
started with.

490
00:33:56 --> 00:34:02
And, my answer is almost,
but not quite.

491
00:34:00 --> 00:34:06
In fact, what will its graph
look like?

492
00:34:04 --> 00:34:10
Well, regardless of what
definition I made for the

493
00:34:09 --> 00:34:15
endpoints of those pink lines,
this function will converge to

494
00:34:16 --> 00:34:22
the following.
From here to here,

495
00:34:19 --> 00:34:25
I'll draw it.
I won't put in minus pi's.

496
00:34:23 --> 00:34:29
I'll leave that to your
imagination.

497
00:34:27 --> 00:34:33
So, there's a hole at the end
here.

498
00:34:33 --> 00:34:39
In other words,
the end of the line is not

499
00:34:36 --> 00:34:42
included.
And, the end of this line,

500
00:34:39 --> 00:34:45
regardless of whether it was
included to start with or not,

501
00:34:43 --> 00:34:49
it's not now.
And here, similarly,

502
00:34:46 --> 00:34:52
I start it here with a hole,
and then go down parallel to

503
00:34:51 --> 00:34:57
the function,
t, slope one.

504
00:34:53 --> 00:34:59
And now, how do I fill in,
so the missing places,

505
00:34:57 --> 00:35:03
this is the point,
pi.

506
00:35:00 --> 00:35:06
This is the point,
negative pi,

507
00:35:02 --> 00:35:08
and there are similar points as
I go out.

508
00:35:05 --> 00:35:11
Well, since the function is
continuous here,

509
00:35:08 --> 00:35:14
the Fourier series will
converge to this orange line.

510
00:35:12 --> 00:35:18
But here, there's a jump
discontinuity,

511
00:35:14 --> 00:35:20
and therefore,
the Fourier series,

512
00:35:17 --> 00:35:23
this function converges to the
midpoint of the jump,

513
00:35:20 --> 00:35:26
in other words,
to here.

514
00:35:22 --> 00:35:28
This function,
in other words,

515
00:35:24 --> 00:35:30
converges to this very
discontinuous looking function,

516
00:35:28 --> 00:35:34
and rather odd how these points
are, I say, but in this case,

517
00:35:33 --> 00:35:39
I can prove to you that it
converges here by calculating

518
00:35:37 --> 00:35:43
it.
Look, this is the point,

519
00:35:41 --> 00:35:47
pi.
What happens when you plug in t

520
00:35:44 --> 00:35:50
equals pi?
You get everyone of these terms

521
00:35:50 --> 00:35:56
is zero, and therefore the sum
is zero.

522
00:35:53 --> 00:35:59
So, it certainly converges,
and it converges to zero.

523
00:36:00 --> 00:36:06
Now, that's a general theorem.
It's rather difficult to prove.

524
00:36:04 --> 00:36:10
You would have to take,
again, an analysis course.

525
00:36:07 --> 00:36:13
But, I don't even get to it in
the analysis course which I

526
00:36:12 --> 00:36:18
teach.
If I had another semester I'd

527
00:36:14 --> 00:36:20
get to it, but I can't get
everything.

528
00:36:17 --> 00:36:23
Anyway, we're not going to get
to it this semester to your

529
00:36:21 --> 00:36:27
infinite relief.
But, you should know the

530
00:36:24 --> 00:36:30
theorem anyway.
People will expect you to know

531
00:36:27 --> 00:36:33
it.
Well, that was half the period,

532
00:36:32 --> 00:36:38
and in the remaining half,
you're going to stay a long

533
00:36:39 --> 00:36:45
time today.
Okay, no, don't panic.

534
00:36:43 --> 00:36:49
I have to extend the Fourier
series.

535
00:36:47 --> 00:36:53
Okay, let me give you the hurry
up version indicating the two

536
00:36:54 --> 00:37:00
ways in which it needs to be
extended.

537
00:37:00 --> 00:37:06
Extension number one --

538
00:37:03 --> 00:37:09

539
00:37:14 --> 00:37:20
The period is not two pi,
but two times,

540
00:37:18 --> 00:37:24
I'll keep the two just to make
the formulas look as similar as

541
00:37:24 --> 00:37:30
possible to the old ones.
The period, let's say,

542
00:37:29 --> 00:37:35
instead of two pi,
is two times L.

543
00:37:34 --> 00:37:40
Now, I think you know enough
mathematics by this point to

544
00:37:37 --> 00:37:43
sort of, I hope you can sort of
shrug and say,

545
00:37:40 --> 00:37:46
well, you know,
isn't that just kind of like

546
00:37:42 --> 00:37:48
changing the units on the
t-axis?

547
00:37:44 --> 00:37:50
You're just stretching.
Yeah, right.

548
00:37:46 --> 00:37:52
All you do is make a change of
variable.

549
00:37:49 --> 00:37:55
Now, should we make it nicely?
I think I'll give you the final

550
00:37:52 --> 00:37:58
answer, and then I'll try to
decide while I'm writing it down

551
00:37:56 --> 00:38:02
how much I'll try to make the
argument.

552
00:38:00 --> 00:38:06
First of all,
the main thing to get is,

553
00:38:04 --> 00:38:10
if the period is not pi but L,
what are the natural versions

554
00:38:11 --> 00:38:17
of the cosine and sine to use?
Use the natural functions.

555
00:38:18 --> 00:38:24
Natural has no meaning,
but it's psychologically

556
00:38:23 --> 00:38:29
important.
In other words,

557
00:38:26 --> 00:38:32
what kind of function should
replace that?

558
00:38:33 --> 00:38:39
I'll certainly have a t here.
What do I put in front?

559
00:38:37 --> 00:38:43
I'll keep the n also.
The question is,

560
00:38:40 --> 00:38:46
what do I fix?
What should I put here in

561
00:38:43 --> 00:38:49
between in order to make the
thing come out,

562
00:38:47 --> 00:38:53
so that it has period 2L?
You probably should learn to do

563
00:38:52 --> 00:38:58
this formally as well as just
sort of psyching it out,

564
00:38:56 --> 00:39:02
and taking a guess,
or memorizing the answer.

565
00:39:00 --> 00:39:06
If this is the t-axis,
here is t and L,

566
00:39:03 --> 00:39:09
zero and L.
What you want to do is make a

567
00:39:08 --> 00:39:14
change of variable to the u-axis
where the axis is the same.

568
00:39:14 --> 00:39:20
This is still the point.
But, L, now,

569
00:39:17 --> 00:39:23
on the u coordinate,
has the name pi.

570
00:39:21 --> 00:39:27
Now, so I'm just describing a
change of variable on the axis.

571
00:39:26 --> 00:39:32
What's the one that does this?
Well, when t is L,

572
00:39:31 --> 00:39:37
u should be pi.
So, t should be L over pi.

573
00:39:37 --> 00:39:43
When u is pi,

574
00:39:39 --> 00:39:45
t is L, and vice versa.
How about expressing u in

575
00:39:43 --> 00:39:49
terms, well, then u is equal to
pi over L times t.

576
00:39:49 --> 00:39:55
That's the backwards form of

577
00:39:52 --> 00:39:58
writing it, or the forward form,
depending upon how you like to

578
00:39:58 --> 00:40:04
think of these things.
Okay, so the cosine should be

579
00:40:05 --> 00:40:11
pi over L times t,
in order that when t be L,

580
00:40:10 --> 00:40:16
it should be like cosine of n
pi,

581
00:40:16 --> 00:40:22
which is what we would have
had.

582
00:40:19 --> 00:40:25
So, if t is equal to L,
in other words,

583
00:40:25 --> 00:40:31
where is this from?
What am I trying to say?

584
00:40:32 --> 00:40:38
That's the function.
This one is probably a little

585
00:40:36 --> 00:40:42
easier to see.
Where is this one zero?

586
00:40:40 --> 00:40:46
The sine functions that we used
before was zero at zero pi,

587
00:40:46 --> 00:40:52
two pi, three pi.
Where is this one zero?

588
00:40:50 --> 00:40:56
It's zero at zero.
When t is equal to L,

589
00:40:54 --> 00:41:00
it's zero.
When t is equal to 2L,

590
00:40:58 --> 00:41:04
so, this is the right thing.

591
00:41:03 --> 00:41:09
So, it's zero.
It's periodic,

592
00:41:05 --> 00:41:11
and it's zero plus or minus L
plus or minus 2L.

593
00:41:08 --> 00:41:14
And, in fact,
formally you can verify that

594
00:41:11 --> 00:41:17
it's periodic with period 2L.
So, in other words,

595
00:41:15 --> 00:41:21
we want a Fourier expansion to
use these functions as the

596
00:41:19 --> 00:41:25
natural analog of what would be
up there.

597
00:41:22 --> 00:41:28
So, the period of our function
is 2L, and the formula is,

598
00:41:26 --> 00:41:32
I'll give you the formula.
It's f of t equals

599
00:41:32 --> 00:41:38
identical summation,
an, except you'll use these as

600
00:41:38 --> 00:41:44
the natural functions instead of
cosine nt and sine

601
00:41:45 --> 00:41:51
nt. So, n pi t over L

602
00:41:49 --> 00:41:55
plus bn, okay,
I'm tired, but I'll put it in

603
00:41:54 --> 00:42:00
anyway, n pi t over L.

604
00:42:00 --> 00:42:06
Yeah, but of course,
what about the formulas for an?

605
00:42:04 --> 00:42:10
Somebody up there is watching
over us.

606
00:42:08 --> 00:42:14
Here are the formulas.
They are exactly what you would

607
00:42:13 --> 00:42:19
guess if somebody said produce
the formulas in ten seconds,

608
00:42:19 --> 00:42:25
and you'd better be right,
and you didn't have time to

609
00:42:24 --> 00:42:30
calculate.
You say, well,

610
00:42:27 --> 00:42:33
it must be, let's do the cosine
series.

611
00:42:32 --> 00:42:38
Okay, let's not do a cosine.
So, it's one over L

612
00:42:36 --> 00:42:42
times the integral from negative
L, in other words,

613
00:42:41 --> 00:42:47
wherever you see an L,
wherever you see a pi,

614
00:42:45 --> 00:42:51
just put an L times the f of t
cosine, and now we'll use our

615
00:42:50 --> 00:42:56
new function,
not the old one.

616
00:42:52 --> 00:42:58
I submit that's an easy,
if you know the first formula,

617
00:42:57 --> 00:43:03
then this would be an easy one
to remember.

618
00:43:01 --> 00:43:07
All you do is change pi to L
everywhere.

619
00:43:06 --> 00:43:12
Except, you got to remember
this part.

620
00:43:09 --> 00:43:15
Make it a function periodic of
period 2L, not 2pi.

621
00:43:15 --> 00:43:21
And similarly,
bn is similar.

622
00:43:18 --> 00:43:24
It looks just the same way.
And, how about,

623
00:43:22 --> 00:43:28
and the same even-odd business
goes, too, so that if f of t,

624
00:43:28 --> 00:43:34
for example,
is even, and has period 2L,

625
00:43:33 --> 00:43:39
then the function,
then the best formula for the

626
00:43:38 --> 00:43:44
an will not be that one.
It will be two over L,

627
00:43:45 --> 00:43:51
and where you integrate only
from zero to L,

628
00:43:49 --> 00:43:55
f of t cosine.

629
00:43:51 --> 00:43:57

630
00:44:01 --> 00:44:07
So, now, the bn's will be zero,
and you'll just have positive,

631
00:44:07 --> 00:44:13
etc.
for L.

632
00:44:08 --> 00:44:14
As I say, this is important
case, particularly if the period

633
00:44:13 --> 00:44:19
is two, in other words,
if the half period is one

634
00:44:18 --> 00:44:24
because in the literature,
frequently one is used as the

635
00:44:24 --> 00:44:30
standard normal reference,
not pi.

636
00:44:27 --> 00:44:33
Pi is convenient mathematically
because it makes the cosines and

637
00:44:33 --> 00:44:39
sines look simple.
But, in actual calculation,

638
00:44:39 --> 00:44:45
it tends to be where L is one.
So, usually you have a pi here.

639
00:44:45 --> 00:44:51
You don't have just nt.
Well, I should do a

640
00:44:49 --> 00:44:55
calculation, but instead of
doing that, let me give you the

641
00:44:55 --> 00:45:01
other extension.
Fortunately,

642
00:44:58 --> 00:45:04
there are plenty of
calculations in your book.

643
00:45:04 --> 00:45:10
So, let me give you in the last
couple of minutes the other

644
00:45:10 --> 00:45:16
extension.
This is going to be a very

645
00:45:14 --> 00:45:20
important one for us next time.
Typically, in applications,

646
00:45:21 --> 00:45:27
well, I mean,
the first thing,

647
00:45:24 --> 00:45:30
periodic functions are nice,
but let's face it.

648
00:45:30 --> 00:45:36
Most functions aren't periodic,
I have to agree.

649
00:45:37 --> 00:45:43
So, all this theory is just
about periodic functions?

650
00:45:40 --> 00:45:46
No.
It's about functions.

651
00:45:42 --> 00:45:48
Really, it's about functions
where the interval on which you

652
00:45:46 --> 00:45:52
are interested in them is
finite.

653
00:45:48 --> 00:45:54
It's a finite interval,
not functions which go to

654
00:45:52 --> 00:45:58
infinity.
For those, you will have to use

655
00:45:54 --> 00:46:00
Fourier transforms,
Fourier transforms,

656
00:45:57 --> 00:46:03
not Fourier series.
But, if you are interested in a

657
00:46:02 --> 00:46:08
function on a finite interval,
then you can use Fourier series

658
00:46:06 --> 00:46:12
even though the function isn't
periodic because you can make it

659
00:46:11 --> 00:46:17
periodic.
So, what you do is,

660
00:46:13 --> 00:46:19
if f of t is on,
let's take the interval from

661
00:46:17 --> 00:46:23
zero to L.
That's a sample finite

662
00:46:19 --> 00:46:25
interval.
I can always change the

663
00:46:22 --> 00:46:28
variable to make the interval
from zero to L.

664
00:46:25 --> 00:46:31
I can even make it from zero to
one, but that's a little too

665
00:46:29 --> 00:46:35
special.
It would be a little awkward.

666
00:46:34 --> 00:46:40
So, if a function is defined on
a finite interval,

667
00:46:38 --> 00:46:44
the way to apply the Fourier
series to it is make a periodic

668
00:46:43 --> 00:46:49
extension.
Now, since I have so little

669
00:46:47 --> 00:46:53
time, I'm just going to get away
with murder by just drawing

670
00:46:52 --> 00:46:58
pictures.
So, let me give you a function.

671
00:46:55 --> 00:47:01
Here's my function defined on
zero to L, colored chalk if you

672
00:47:01 --> 00:47:07
please.
Let's make it the function t

673
00:47:05 --> 00:47:11
squared,
and let's make L equal to one.

674
00:47:09 --> 00:47:15
That function is not periodic.
If I let it go off,

675
00:47:13 --> 00:47:19
it would just go off to
infinity and never repeat its

676
00:47:17 --> 00:47:23
values, except on the left-hand
side.

677
00:47:20 --> 00:47:26
But, I'm not even going to let
it be on the left hand side.

678
00:47:25 --> 00:47:31
It's only defined from zero to
one as far as I'm concerned.

679
00:47:29 --> 00:47:35
Okay, that function has an even
periodic extension.

680
00:47:35 --> 00:47:41
And, its graph looks like this
extended to be an even function.

681
00:47:39 --> 00:47:45
Okay, now, that means from zero
to negative L,

682
00:47:44 --> 00:47:50
you've got to make it look
exactly as it looked on the

683
00:47:48 --> 00:47:54
right-hand side.
Otherwise, it would be even.

684
00:47:51 --> 00:47:57
And now, what do I do?
Well, now I've got,

685
00:47:54 --> 00:48:00
from minus L to L.
So, all I'm allowed to do is

686
00:47:59 --> 00:48:05
keep repeating the values.
In other words,

687
00:48:03 --> 00:48:09
apply the theory of Fourier
series to this guy,

688
00:48:06 --> 00:48:12
use a cosine series because
it's an even function,

689
00:48:10 --> 00:48:16
and then everything you want to
do, you say, okay,

690
00:48:14 --> 00:48:20
all the rest of this is
garbage.

691
00:48:16 --> 00:48:22
I only really care about it
from here to here.

692
00:48:20 --> 00:48:26
And, that's what you will plug
into your differential equation

693
00:48:24 --> 00:48:30
on the right-hand side,
just that part of it,

694
00:48:28 --> 00:48:34
just this part of it.
How about the odd extension?

695
00:48:33 --> 00:48:39
What would that look like?
Okay, the odd extension,

696
00:48:37 --> 00:48:43
here I start like this.
And now, to extend it to be an

697
00:48:41 --> 00:48:47
odd function,
I have to make it go down in

698
00:48:44 --> 00:48:50
exactly the same way it went up.
And, what do I do here?

699
00:48:49 --> 00:48:55
I have to make it start
repeating its values so it will

700
00:48:53 --> 00:48:59
look like this.
So, the odd extension is going

701
00:48:57 --> 00:49:03
to be discontinuous in this
case.

702
00:49:01 --> 00:49:07
And, what's the Fourier series
going to converge to?

703
00:49:05 --> 00:49:11
Well, in each case,
to the average,

704
00:49:07 --> 00:49:13
to the midpoint of the jump,
and the odd extension looks

705
00:49:12 --> 00:49:18
like this, and this will give me
assigned series.

706
00:49:16 --> 00:49:22
Okay, you've got lots of
problems to do.